Optimal. Leaf size=53 \[ -\frac {b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2591, 14} \[ -\frac {b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2591
Rubi steps
\begin {align*} \int \csc ^4(e+f x) (b \tan (e+f x))^n \, dx &=\frac {b \operatorname {Subst}\left (\int x^{-4+n} \left (b^2+x^2\right ) \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {b \operatorname {Subst}\left (\int \left (b^2 x^{-4+n}+x^{-2+n}\right ) \, dx,x,b \tan (e+f x)\right )}{f}\\ &=-\frac {b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 46, normalized size = 0.87 \[ \frac {b \csc ^2(e+f x) (\cos (2 (e+f x))+n-2) (b \tan (e+f x))^{n-1}}{f (n-3) (n-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 86, normalized size = 1.62 \[ \frac {{\left (2 \, \cos \left (f x + e\right )^{3} + {\left (n - 3\right )} \cos \left (f x + e\right )\right )} \left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n}}{{\left (f n^{2} - {\left (f n^{2} - 4 \, f n + 3 \, f\right )} \cos \left (f x + e\right )^{2} - 4 \, f n + 3 \, f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.48, size = 13019, normalized size = 245.64 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 55, normalized size = 1.04 \[ \frac {\frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 1\right )} \tan \left (f x + e\right )} + \frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 3\right )} \tan \left (f x + e\right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 138, normalized size = 2.60 \[ -\frac {2\,{\left (-\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{2\,{\sin \left (e+f\,x\right )}^2-2}\right )}^n\,\left (9\,\sin \left (2\,e+2\,f\,x\right )-6\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )-4\,n\,\sin \left (2\,e+2\,f\,x\right )+2\,n\,\sin \left (4\,e+4\,f\,x\right )\right )}{f\,\left (30\,{\sin \left (e+f\,x\right )}^2-12\,{\sin \left (2\,e+2\,f\,x\right )}^2+2\,{\sin \left (3\,e+3\,f\,x\right )}^2\right )\,\left (n^2-4\,n+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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